Odds are good that your cable company delivers all video in the native rate it was shot in. So, standard definition channels are shot, and broadcast, at 480i. Your HD shows are either going to be 720p or 1080i (not 1080p) video.
Nothing that is broadcast will be a perfect match for your 1080p display, but 1080i broadcasts should be upconverted very well if you have a quality 1080p display. Some people go cheap, and the quality of their 1080p display is 'lacking' to say the least.
For true 1080p: HD DVD or Blu-ray is the only way to do this right now. DVD is NOT HD. It is 480i video.
There are numerous articles about what interlaced video is vs. progressive scan video. I just don't have any links handy right now on the subject.